# The Universality of the Radon Transform

## Leon Ehrenpreis

### Abstract

Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elabo ... More

Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elaborates on them and puts them in a general framework.

*Keywords: *
delta functions,
hyperplanes,
nonparametric,
geometric objects,
differential equations,
F. John

### Bibliographic Information

Print publication date: 2003 |
Print ISBN-13: 9780198509783 |

Published to Oxford Scholarship Online: September 2007 |
DOI:10.1093/acprof:oso/9780198509783.001.0001 |

### Authors

#### Affiliations are at time of print publication.

Leon Ehrenpreis, *author*

Professor of Mathematics, Temple University, USA

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